Time: A Guide for Travellers

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This year is the centenary of the birth of the great logician Kurt Gödel. One of his greatest discoveries had nothing to do with mathematical logic. It was that Einstein’s theory of relativity allows universes in which time travel is possible. Here we describe Gödel’s strange universe and discuss the history of the idea of time travel to the future and to the past in science fiction and science fact. In particular we look at the traditional story line of ‘back to the future’ and show how it is possible to be part of the past but not to change the past, and what physicists think about time travel in the light of quantum theory.

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Professor John Barrow


One of the reasons I chose to talk about time travel today is because 2006 is the centenary of the birth of Kurt Gödel, who was the greatest of all magicians, and in 2006, there will be many events and conferences around the world celebrating the achievements of Kurt Gödel, who died in the 1970s.

Although he’s very famous for his work in logic, and theorems of arithmetic, he is also famous in cosmology and astronomy, because he was someone who discovered that Einstein’s Theory of General Relativity allowed time travel to occur. He found the solution of Einstein’s equation, which described a universe in which time travel to the past was possible.

This shocked Einstein very much. Einstein and Gödel used to spend their lunchtimes in Princeton walking and talking together. They used to talk much about problems of time and space-time structure. Gödel shocked Einstein by finding a solution of Einstein’s theory which described a rotating universe in which time travel was possible. The reason he was so shocked, I think, is that, like other people, he thought that time travel would somehow be in contradiction with the other laws of physics, but here was a completely self-consistent theory of physics that allowed time travel to occur without any other disruption to things.

The reason this sort of thing is possible, in general relativity, is because what it tells us about the universe is that space and time are not absolutely fixed and flat; particles and objects and planets just move around on the stage like actors following particular rules, but the stage itself is changeable. It’s like a rubber sheet or a trampoline: as objects move around, they change the geometry and the shape of the space on which they move, and they change the rate at which time flows on their clocks compared with clocks of people far away. So if you have a small object, like a planet such as Earth, sitting on space and time, it only produces a small indentation, as it were, to the flat simple geometry you would expect. But you could have much more contorted and violent things happen, rather like twisting the space over to produce an “S” shape, and then you can have situations where a very complicated thing has happened to the flow of time. The recipe for getting this to happen, people have found, following Gödel’s lead, is very much to have rotation present in the universe or the space.

Rotation has peculiar effects in Einstein’s theory. In Newton’s theory, if we spin a top over here, it doesn’t have any effect on you over there, just because it’s spinning. There’s a tiny gravitational effect between you, but the spin doesn’t have any significant effect. But if you have a picture of space and it’s rather like a great rubber trampoline and you put an object in space over here, it produces an indentation - if we start twisting it and spinning it, the twist to the trampoline starts to communicate itself to things further away, and they get moved in the same sense that the top is spinning. This is called the dragging inertia and we think it’s now observed in space, both from the effect of the rotation of the Earth and in other star systems. Rotation is a complicated and unusual effect in the General Theory of Relativity, and, as in Dali’s famus soft watch image, is very much the image that you should have about the way time can flow at different rates in different places, depending on how much mass there is in different places, and how much distortion there is to the shape and the geometry of space.

Gödel’s famous universe is usually known as Gödel’s universe, and it is a rotating universe. It can’t describe our universe, because it’s not an expanding universe, but it could be that the essential ingredients present in Gödel’s universe that create the time travel possibility might still be possible in other rotating universes that do expand, and that’s the great interest in this model.

Some years ago, Frank Dibler, who was a collaborator of mine, produced a recipe for a time machine created from Gödel’s type of recipe, a model. Again, it involves rotation, and it’s rather unrealistic for us to construct, but imagine that you had a great solid iron cylinder that’s 100km in length, and it’s about 10km across, and it rotates very, very rapidly – hundreds and hundreds of times per second. It’s rather like a neutron star. A neutron star may be rotating 500 times a second, and it’s about the size of London, but it has the mass of the Sun. Astronomically, there are bizarre objects. What you would do is that if you flew in in a spacecraft to the very close environment of a huge rotating cylinder, then if you flew around the cylinder, you would experience time travel into the past - so the more laps you do of the cylinder before returning home, the further into the past you will find yourself arriving back when you get home. All the time, your watch is going forwards, and you are recording the steady fall and flow of time, but you will nonetheless be able to fly back to where you began and arrive in the past of the people who you meet when you return. We know that in Einstein’s theory this is a possibility. There are solutions of Einstein’s theory that allow time travel into the past. The issue is whether there are other restrictions, other rules, that you have to use to sift out solutions of Einstein’s theory which rule these out, so that they are allowed but they don’t occur in reality. It may be rather like balancing a pin on its point. This is something that could remain for ever, in principle, but in practice, it gets perturbed and it falls down, so it’s unstable.

People’s objections to these situations of travelling in time into the past revolve around a number of awkward supposed paradoxes, and I want to show you what some of those are, or perhaps refresh your memory about some of them. The first usually goes by the name of the cumulative audience paradox, or some people just call it the tourist from the future problem. If it’s possible to travel backwards in time, then you would expect that there would be certain events that people might like to visit to check out for themselves. So certain events in, for example, the Christian religious tradition, like the resurrection or the crucifixion or the birth of Christ at Christmas, these events would be extremely crowded, with millions and millions of visitors! Lectures like this one would be very crowded! Some people worry about this, and it’s not clear what to make of it. Maybe it is why the underground is always so crowded when I come to London ! Some people would argue that such visitors are here, and I’ve even, in America, encountered conferences, all of the speakers at which claim to be people who have travelled back from the future or come from other planets. It’s comforting to know that such conferences take place. Some people say, “Oh this is a crazy idea!” I say, on the contrary, it’s very good to have all those people in one place at the one time, because you know exactly where they are, so they are not flying your aeroplane or driving your car!

Let’s suppose you think that there are not lots of time travellers around, either for the reason I’ve just said, or you just think there is no evidence for it, so time travel is not obvious and it’s not occurring all the time. People are not popping in from the future like Dr Who. What reasons might you think up for that state of affairs? It could be that although time travel is possible, it requires super-advanced technology, and nowhere in the universe is it yet possible, so there are no time travellers yet – but there may well be one day. In a sense, this means that the absence of evidence is not evidence of absence; it is just not relevant. The second possibility is that time travel is extremely dangerous, so people who develop this capacity either don’t survive long enough to come back to the past, or something even more catastrophic happens and they actually render themselves extinct.

So creating possibility of time travel is rather like the precipitating a total environmental disaster in space and time. It could be that time travel is possible, easy to do if you are very advanced, and happens all the time, but nobody visits us, because we are too uninteresting, we are too boring. There are millions of civilisations around the universe, rather like our own, that had approximately a similar history over billions of years, and there is no particular reason to come and see us. It is a bit like UFOs: it’s difficult to believe that people fly a million light years across the galaxy to land in a potato field in East Anglia and then take off and go home! There has to be a reason for coming.

The other possibility is completely the opposite: that we are too interesting, and there is a policy of non-interference, that very advanced, highly enlightened beings don’t wish to return the sequence of events in time that we experience, because there are bad consequences of doing that. It could be it is a rather exclusive club, and you are only, as it were, inducted into this club, it’s only revealed to you how to do this, how to take part in time travel, if you have shown yourself to be an extremely responsible and trustworthy civilisation, and maybe we have not done that.

But much more likely is that it is just too expensive. If you worry about building Channel Tunnels or extra runways and so forth because of the expense and long term investment, it is likely to be prohibitively expensive, to divert resources which in advanced civilisations are going to be required to deal with all sorts of other problems that we can foresee – environmental ones, the running out of natural resources and fuel and energy and coping with disease and so on. So there are not dramatic reasons why we might indeed discover that it is entirely reasonable that even if time travel is possible, we do not see any time travellers.

Another feature I like about time travel is that one of the best pieces of evidence against it is the fact that non-zero interest rates exist. You can see that if, in 3003, you had a Pound, if such things existed – probably something rather more European or international – and if you could travel back to 2003 and pop your £1 in the bank, say interest rates are 4% for a thousand years, you pop along to the bank, and you have got £108 million billion in your savings bank account, just enough to get your British Rail ticket back home in 3003! So you can see what the effect of this would be, that if there were investors from the future coming, picking stocks that they knew were going to produce big returns in the next few years, the effect would have to drive interest rates to zero. Similarly, if rates were negative, you could travel to the future to purchase something which is going to be successful, and then, when it’s cheaper, come back here and re-sell it at the higher price. So either way, positive or negative interest rates would be driven to zero by time travellers. Bankers should keep an eye out for that!

The most famous type of time travel paradox is usually called the grandmother paradox. It is simply the idea that you might be able to produce a factual contradiction. So if you can travel backwards in time, you could, for example, kill yourself, or your mother or your grandmother, which is the original idea of the story, at some time in the past, and if you kill your mother, you prevent your own birth, and this appears then to be a factual contradiction with your own existence now. We will return to that in a moment.

The other rather nice paradox is what I call the information paradox. Suppose that I read Shakespeare’s Hamlet, I know all the words, look up my notes, and I travel back into the past and I meet young William Shakespeare in a farmyard somewhere in Stratford when he was a teenager and I tell him about this wonderful idea of the play and give him all the lines and he writes them all down. Where did the play Hamlet come from? He learns it from me, but I learnt it from him! So we seem to have created information out of nothing by having this loop in time.

Well, all these supposed paradoxes are really not true paradoxes. They are of course the basis for all those science fiction movies of the Back to the Future sort. Everybody in those stories goes back to the past and does something which changes something that happens now. The message here is that you can participate in the past, but you cannot change it. The past is what it was. If a book was printed with certain words in it in 1066, it was printed. There is not some other past that you can go back and create. If you participated and did something in the past, it would be part of the historic record. A good way to think of it is to imagine that you have a river. It’s a familiar image, to think of time rather like a flowing river going forwards. But the possibility of time travel into the past is rather like taking a tributary off the river, taking a little channel, running it backwards, perhaps having a little pump, and diverting the stream and putting it back into the river elsewhere. This is like time travel. The water that comes back round participates in the river, but there is only one river. What you end up with is a single self-consistent flow of water that incorporates the extra channel and the main flow. There are not two flows of water, one with and one without the channel added.

Here’s another way of looking at it: if time is linear and goes forward, it’s rather like having a collection of soldiers, and they just march one behind the other. What is unambiguous about the situation is that everyone can say that any other soldier is either in front of them or behind them. The future and the past, as it were, are unambiguous for each person. But suppose the soldiers now march in a circle. The situation is completely changed. Everybody has everybody else both in front of them and behind them. This is the situation or analogy that exists when you have time travel, when the path of time is closed, and what you have to ask and look for is just what is the self-consistent pattern of marchers in their circle, what is the self-consistent history that can take place in a loop? There is no going back into the past to change things, to add something extra. There is one historical story, which is a closed path.

An example I like to give of what those self-consistent histories would look like: suppose I’m going to travel backwards in time, and I’m planning to kill myself as a baby in order to produce a factual contradiction in the experience of the universe. So I travel backwards in time, 53 years ago. I see my mother holding myself as a little baby in her arms, and I have a gun with me and I’m going to shoot the baby. I pick up my rifle and I take aim and I go to pull the trigger, but just as I pull the trigger, I feel a spasm in my shoulder from an old injury, and it makes the gun jump, it goes bang, the bullet misses, but the bang frightens my mother, who drops the baby, who injures his shoulder…! That’s a self-consistent history, and we have to require in classical physics that all histories which can actually occur logically have that self-consistent form.

If you go on to quantum theory, then there are other possibilities. One of the things that you have to recognise about physics is that the laws of physics allow many things to happen which will never be seen. They are never seen not because they are forbidden by the laws of physics, but the conditions which enable them to occur are so fantastically improbable that they just never arise. If I dropped this glass on the floor and smashed it into pieces, this is a very common accidental event that we see very frequently. The laws of physics allow the time reverse of that process to occur. It does defy any conservation law, so lots of little fragments of glass coming together spontaneously to make a glass in a very organised state – this is allowed by the laws of physics, but we never see it happening, because the conditions required for that event to occur, the starting conditions, are fantastically improbable. Every little splinter of glass would have to be moving exactly the right way to come together at the same moment to make the glass, and that is so improbable that we will never witness it in our entire lifetime. Time travel may well be like this. It may be allowed by the quantum laws of physics for example, but the conditions required for it actually to occur spontaneously are so improbable that we never witness it.

This table here is full of lots of little molecules all jiggling around in random ways, but if all the molecules happened to be moving upwards at the same moment, the table would levitate and move into the air. This would not defy the laws of physics, but in practice, it is so improbable that we never see it, and if someone tells you they have seen it, it is much more likely that they are deluded than that it has actually occurred. You have to bear in mind that just because something is possible does not mean that it is going to occur.

Another consideration we would have to make, which is really the same one, is that if you have loops in time, and if you follow the loop, you eventually double back and repeat yourself. It is most probable that systems evolve from a state of some disorder into more disorder. Things tend to go from bad to worse. There are just so many more ways in which an orderly system can become disorderly than vice versa. This is what we tend to see on the average in practice. You can see that this is rather awkward in the context of time travel because if the entropy, the disorder is going to increase all the time, if you want to have a closed time path, eventually you come back with a rather awkward junction condition. So you need very unusual, highly improbable configurations, which always maintain the same degree of overall order, for them to be recurrent. They can be almost recurrent, so they can be almost the same. So you’ll be quite happy to return as yourself, as it were, but with say just one atom missing, or one or two atoms changed.

So it could be that time travel really is impossible, that there is some as yet unfound principle that excludes all these solutions of Einstein’s Theory and says they do not apply to our world. They only apply to worlds which have negative density or some other strange properties. Or it could mean that it is very improbable. In some sense, we know this is the case, because we do not see it happening all the time. Another version of this is that it is unstable, so that if you create the conditions for time travel, it’s rather like balancing this pointer: it falls and the conditions change, so that you have a terrible job trying to maintain a situation which would allow time travel.

Now, instabilities are not always bad. I recall learning once that some modern fighter aircraft are deliberately made to be aerodynamically unstable because it allows them to be manoeuvred much more rapidly, so they are already, as it were, on the verge of changing their configuration because of instability before the manoeuvre takes place.

There is another option that my colleague Stephen Hawking probably subscribes to:

he might actually think that time travel is only possible at the beginning of the universe, and of course then there’s no past to travel into. So you could produce a conjecture that physicists might one day be able to prove, that you cannot build a time machine except at the beginning of the universe - when there is no past to travel into. You can travel into the future, by building a time machine, and you just need to organise matter in such a way that these unusual closed paths are possible.

At a more serious scientific level, people take an interest in time travel because the quantum picture of the world is rather different from the classical one. In the quantum picture, if you want to ask: if you do something here, what happens in the future? In Newton’s theory, that’s rather simple: if you accelerate something, it will move in a particular way and it will have a particular location in the future. Quantum theory is rather different. What you ask of the theory is that if you have a particular arrangement of matter here and now, what is the probability that when you measure the system again, you observe it again, in the future, you will get a particular result, a particular configuration? There is not a unique configuration that will be seen in the future. There is a most probable one, but there are others which are less probable. This is something that we see in the laboratory all the time, with electrons and atomic systems. A system may be 50% found to be in one state and 50% to be another state. One way of interpreting what is going on there is that in some sense the system takes all possibly histories that it could take into the future, and on the average, one of them tends to dominate and gives you the probabilities that you measure in your experiment. That interpretation gives you a wonderfully accurate answer, which you can test by experiment, and the tests really are fantastically accurate, more accurate than anything else that we know about anything. So the predictions are correct to 16 decimal places.

Some scientists had wondered when you carry out calculations, when they develop into rather ordinary experimental procedures like this, and you take the picture of what’s the average chance of it happening, what the average answer turns out to be depends on what are all the possibilities that you take the average over. If you included in the possibilities some of them which involved travelling in time, you might get a different answer from the thing that you are going to go and measure than if you excluded them. So it can be that you can end up with different predictions about experimentally measurable things if time travelling paths exist in the universe but you don’t need to observe them. They just have an indirect effect on other quantum phenomena.

After one of my earlier talks, someone in the audience asked me by email whether there could be other times, and so I replied and said I would try and mention this in this talk, because it is going to be about time. Time travel is also perhaps connected with the fact as to whether there really is just one dimension of time, or whether there may be more than one dimension of time. The currently favoured theory, that we might call “Theory of Everything,” which I will talk about in February in this series, is so-called string theory. String theory is the theory of space and of time and of the most elementary states of matter. Instead of thinking of the most elementary bits of matter as just little point particles, it views them as little lines or little loops of energy, and those loops have a certain elasticity, as it were, rather like little rubber bands, and the elasticity depends on the temperature of the world in which they exist. So if the temperature is very low, like in this room, the elasticity is very strong, and the loops zap down and become more and more like points. In our relatively low energy, low temperature world, all the little stringy loops just behave like point particles, and you explain all the observations of the world that we see based on the earlier assumption that they really were point particles. But when you go to fantastically high energies, of the sort you would encounter in the distant past of the universe, or perhaps near the centre of the most energetic objects, then elasticity becomes low, and the points become intrinsically stringy and loop-like, and they behave in a different way. Like any string, whether it is a guitar string, or string that you are strumming on your finger, it has energies of vibration, and those natural energies of vibration of the strings, using E=MC 2, Einstein’s famous formula, correspond to particular masses, and they are the masses of what we call elementary particles. The challenge of this string theory is to extract mathematically those predictions. Hidden in the theory is the information about why all the elementary particles we see have the masses that they do, because these masses correspond to excitations of the stringy energy.

The strange thing about these theories is that they have lead us to all sorts of other wonderful properties if and only if there are many more dimensions of space time than the three of space and one of time that we see. In particular, there need to be ten dimensions of space and time in total for these theories to have their beautiful properties. Try and add it as three space and one time, and you get infinite answers to all sorts of questions that you ask in the theory. It was a great discovery by Mike Green in Cambridge, in our Department, and Jonathan Schotts at Caltech, to demonstrate this in 1982, that there was this one and only one class of theories of everything that had this property that they were finite and all calculations could be done in them.

The way this is generally interpreted is that one of these dimensions is time, three of the dimensions are the dimensions of space we walk around in, and they are big. The remaining dimensions are also dimensions of space, but they are fantastically and imperceptibly small, and you have to try and find ways to detect their presence in elementary particle experiments. Their size would perhaps only be as small as 10 to the minus 33 of a centimetre, but they could be as large as about a thousandth of a millimetre, which is something that you could almost see. However, the peculiarity of this theory is that it has this beautiful property of finiteness and so forth if the numbers of dimensions of space plus time is ten, but the theory is actually completely silent on how many of those ten need to be times and how many have to be spaces. Physicists just assume that one of them is a time and the others are all dimensions of space, but you are not forced to do that, you are not required to do it, and the theory would have all the same beautiful mathematical properties if there were 2 times and 8 spaces, or 3 times and 7 spaces, 9 times and one space. It wouldn’t look much like our world. But this is rather puzzling, so people wonder how is the split chosen? We don’t know why 3 dimensions have become big and the others have remained small, whether that was a random effect or whether there is some deep principle which means it has to be 3, and we don’t know why only one of them appears to have set up as a time. Perhaps there are other times, but like the other dimensions of space, they are fantastically small in the sense of being fleeting and short lived, and we don’t see them in any way.

If you imagine that the world could have popped out with any number of dimensions of space and time, there is a rather interesting consideration. Is there some situation in which life and observers and astronomers and sting theorists can exist and other situations where they cannot? The answer is rather remarkable. Suppose you imagine that there could be any number of dimensions of time on a graph – so one, 2, 3, 4, 5 and so on – and any number of dimensions of space. We are where there will be one dimension of time and 3 dimensions of space. You could then start to ask what would be the characteristics of being in worlds with more times, more dimensions of space and so on, and the answer is rather striking. If we stick to worlds of one dimension of time, then if we have any other bigger number of dimensions, bigger than 3, there can be no atoms, no planets, no stable structures of any sort, so there are no forces which can exist which bind things together. So we could not expect ourselves to be existing in a world with 4 or 5 or 6 or 7 dimensions of space. There could be no atoms, no gravitational binding to planets and stars; it is a completely uninteresting and dead universe that has more than 3 space dimensions. Similarly, if you have very small numbers of dimensions of space, just 2, then things are too simple to give you the structures that you want. There cannot be any gravitational forces, and there is no interesting structure at all.

If you start to think about worlds where there are more times, the 3-dimensional case is one of space, rather uninteresting – everything moves faster than light, there are no interesting structures. If you try to have more times, with one space, again, you have the mirror image of the situation: there are no atoms, there are no bound structures, you cannot do anything complicated. The more interesting part is to say if we had say 4 dimensions of space and 6 of time, or 5 of time and 5 of space, that string theory allows, those worlds have very strange properties which arise through the way mathematical equations governing waves and changes behave in those worlds, that the future is not determined by the present. So there is complete unpredictability. The future is only partly determined by the present. So you are not able to have an intricate and complicated structure, where if you do something now, you add a cause, you get a predictable effect. Having these extra time dimensions and extra space dimensions produces a situation where very fine-tuned complex structures and the ability to produce something like a brain is really undermined. You have an unpredictable and somewhat chaotic type of world.

If you look more carefully at the situation, say, with 3-dimensional space, like our own, and you think what would happen if you had just 2 times, I call this sort of two-time universe. There are very simple things that you can calculate rather simply. Having 2 times means that there are many more ways for things that change in time to actually change and, in particular, particles which are unstable, neutrons for example, decay much more rapidly because they have more ways to decay. You would find that many forms of matter which are ordinarily stable become unstable. This is interesting because of course you might set about searching for these unusual types of decay in your particle accelerator. So if, for example, a particle decays by a particle going off into another dimension and also into another one that stays in ours, when you looked in the record at the detector, you would see a very peculiar decay which would not be allowed by our three-dimensional laws with one dimension of time, because the conservation of momentum would be guaranteed by the other particle disappearing off into the other time or the other space. So you could systematically search by computer for events that apparently violated the conservation of energy or momentum. So having more than one time, if the time is large and obvious, rather like the time we experience, is very, very awkward. It has all sorts of adverse effects.

Of course you can have, as I said just now, these very, very tiny times, which have very, very restricted existence, just a tiny, tiny fraction of a second, 10 to the minus 43 of a second, and there would in effect be no directly perceptible consequences.

Finally, I was looking on the web the other week, and I came across something very interesting in respect to time travel, which I have missed, but of course if you believe time travel into the past is possible, it does not matter that you have missed it. You cannot miss anything. You could always take part in events in the past. It seems in Perth last spring, on the 31 st of March, they announced “Destination Day.” The idea was that if time travel becomes possible in the future, they were inviting all time travellers to return to Perth on this date, 31 st of March 2005, some 12 noon at their local time, which is GMT plus 8 hours, to have a great rendezvous and celebration. They give you coordinates here, and the actual location. Look at the website, which is www.destinationday.net. I was hoping to show it now. Unfortunately they have taken down the video that they were showing on it. They had some interesting webcams set up on Destination Day at noon to film people who arrived in the town square, who might therefore be potential time travellers from the future. There were a few rather unusual looking pigeons I noticed. There was a chap selling what looked like copies of The Big Issue - it’s good to know that is still in business in the future - and a few other random tourists, about half a dozen people. So it obviously wasn’t a big event, but if you have a look at the website in the future, you may find that they are keeping a continuous update on events there in Western Australia.

I hope I have at least given you some flavour of why people think about time travel still in physics and astronomy. It is a tantalising possibility that is allowed by Einstein’s equations. It is not incompatible with anything else that we know, either in logic or any of the other laws of physics. Physicists and cosmologists suspect that it is something which is allowed in principle, but which is fantastically improbable in practice, because it requires amazingly extreme conditions and very special configurations to effect. But on the other hand, time travel to the future is rather easier. It is a sombreing, though, that if you were around in the 19th century and you asked about time travel to the future, people would have pooh-poohed it in all the same ways that I did just now with regard to time travel in the past, and yet time travel in the future has turned out to be simple to effect and extremely common and routine. Time travel is a live issue in fundamental physics. If you want to make a name, if you could prove that it really was impossible, that there was some fundamental reason for the fact that it cannot exist in our universe, you would become very famous indeed. If you could go in a time machine, you would become even more famous! So I will leave those as homework problems!


© Professor John Barrow, Gresham College, 12 January 2006

This event was on Thu, 12 Jan 2006


Professor John D Barrow FRS

Professor of Astronomy

Professor John D Barrow FRS has been a Professor of Mathematical Sciences at the University of Cambridge since 1999, carrying out research in mathematical physics, with special interest in cosmology, gravitation, particle physics and associated applied mathematics.

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